

Recent Issues 
Volume 10, 7 issues
Volume 10
Issue 7, 1539–1791
Issue 6, 1285–1538
Issue 5, 1017–1284
Issue 4, 757–1015
Issue 3, 513–756
Issue 2, 253–512
Issue 1, 1–252
Volume 9, 8 issues
Volume 9
Issue 8, 1772–2050
Issue 7, 1523–1772
Issue 6, 1285–1522
Issue 5, 1019–1283
Issue 4, 773–1018
Issue 3, 515–772
Issue 2, 259–514
Issue 1, 1–257
Volume 8, 8 issues
Volume 8
Issue 8, 1807–2055
Issue 7, 1541–1805
Issue 6, 1289–1539
Issue 5, 1025–1288
Issue 4, 765–1023
Issue 3, 513–764
Issue 2, 257–511
Issue 1, 1–255
Volume 7, 8 issues
Volume 7
Issue 8, 1713–2027
Issue 7, 1464–1712
Issue 6, 1237–1464
Issue 5, 1027–1236
Issue 4, 771–1026
Issue 3, 529–770
Issue 2, 267–527
Issue 1, 1–266
Volume 6, 8 issues
Volume 6
Issue 8, 1793–2048
Issue 7, 1535–1791
Issue 6, 1243–1533
Issue 5, 1001–1242
Issue 4, 751–1000
Issue 3, 515–750
Issue 2, 257–514
Issue 1, 1–256
Volume 5, 5 issues
Volume 5
Issue 5, 887–1173
Issue 4, 705–885
Issue 3, 423–703
Issue 2, 219–422
Issue 1, 1–218
Volume 4, 5 issues
Volume 4
Issue 5, 639–795
Issue 4, 499–638
Issue 3, 369–497
Issue 2, 191–367
Issue 1, 1–190
Volume 3, 4 issues
Volume 3
Issue 4, 359–489
Issue 3, 227–358
Issue 2, 109–225
Issue 1, 1–108
Volume 2, 3 issues
Volume 2
Issue 3, 261–366
Issue 2, 119–259
Issue 1, 1–81
Volume 1, 3 issues
Volume 1
Issue 3, 267–379
Issue 2, 127–266
Issue 1, 1–126





Abstract

We investigate the Hardy–Schrödinger operator
${L}_{\gamma}=\Delta \gamma \u2215x{}^{2}$ on smooth domains
$\Omega \subset {\mathbb{R}}^{n}$ whose boundaries
contain the singularity
$0$.
We prove a Hopftype result and optimal regularity for variational solutions of
corresponding linear and nonlinear Dirichlet boundary value problems, including the
equation
${L}_{{}_{\gamma}}u={u}^{{2}^{\star}\left(s\right)1}\u2215x{}^{s}$,
where
$\gamma <\frac{1}{4}{n}^{2}$,
$s\in \left[0,2\right)$ and
${2}^{\star}\left(s\right):=2\left(ns\right)\u2215\left(n2\right)$ is
the critical Hardy–Sobolev exponent. We also give a complete description
of the profile of all positive solutions — variational or not — of the
corresponding linear equation on the punctured domain. The value
$\gamma =\frac{1}{4}\left({n}^{2}1\right)$
turns out to be a critical threshold for the operator
${L}_{\gamma}$. When
$\frac{1}{4}\left({n}^{2}1\right)<\gamma <\frac{1}{4}{n}^{2}$, a notion of
Hardy
singular boundary mass ${m}_{\gamma}\left(\Omega \right)$
associated to the operator
${L}_{\gamma}$
can be assigned to any conformally bounded
domain $\Omega $
such that
$0\in \partial \Omega $.
As a byproduct, we give a complete answer to problems of existence of extremals for
Hardy–Sobolev inequalities, and consequently for those of Caffarelli, Kohn and
Nirenberg. These results extend previous contributions by the authors in the case
$\gamma =0$, and by Chern and
Lin for the case
$\gamma <\frac{1}{4}{\left(n2\right)}^{2}$.
More specifically, we show that extremals exist when
$0\le \gamma \le \frac{1}{4}\left({n}^{2}1\right)$ if the mean
curvature of
$\partial \Omega $ at
$0$ is negative. On
the other hand, if
$\frac{1}{4}\left({n}^{2}1\right)<\gamma <\frac{1}{4}{n}^{2}$,
extremals then exist whenever the Hardy singular boundary mass
${m}_{\gamma}\left(\Omega \right)$ of the
domain is positive.

Keywords
Hardy–Schrödinger operator,
Hardysingular boundary mass, Hardy–Sobolev
inequalities, mean curvature

Mathematical Subject Classification 2010
Primary: 35J35, 35J60, 58J05, 35B44

Milestones
Received: 4 January 2016
Revised: 23 February 2017
Accepted: 3 April 2017
Published: 1 July 2017

